WASHINGTON — On the face of it, one of the most powerful pairings in Washington is a hopeless mismatch — a former social worker and liberal Democrat from Baltimore’s working-class Fells Point neighborhood and an old-school, cigar-chomping GOP conservative raised in a dry county in the mountains of southeastern Kentucky.

But in a bitterly divided Congress, the odd couple of Sen. Barbara Mikulski and Rep. Harold Rogers is a rare bipartisan success story.

Mikulski and Rogers are chiefly responsible for divvying up $1 trillion in federal spending as chairwoman of the Senate Appropriations Committee and chairman of the same committee in the House. While their personal backgrounds could hardly be more different, their operating styles are remarkably similar.

Both are pragmatists in a Congress littered with ideologues. Neither minces words or tolerates foolishness. Both prefer deal-making to speechifying. And each understands that in order to strike a deal the other side needs to claim some wins.

“Hal is a conservative but he is not a hard-headed ideologue. He’s a realist,” said former Appropriations Chairman David Obey, D-Wis. “He’s good at saying, ‘Look, I’ll level with you. This is what I can do and this is what I can’t do.’”

Mikulski has a reputation for toughness though her once-fearsome temper seems to have mellowed in recent years. “Her BS quotient is very, very low. She doesn’t tolerate BS and she doesn’t dish it out,” Obey said. “She is very pragmatic, very hard-nosed.”

Rogers and Mikulski face an enormously difficult task: advancing 12 spending bills setting the annual operating budgets for federal agencies and most government programs, ranging from funding the armed forces and overseas military operations to air traffic control, the national parks and forecasting the weather.

House and Senate leaders used to give great deference to the committees but their standing has slipped of late. Senate Majority Leader Harry Reid, D-Nev., was a longtime Appropriations Committee insider but has usually given short shrift to the panel’s pleas for floor time to debate its bills. House Speaker John Boehner, R-Ohio, barely hid his disdain for the panel’s parochial, clubby ways during his first two decades in the House. One of his first acts as speaker was to impose a ban on popular home-district pet projects known as earmarks.

The result is that many lawmakers have less reason to vote for spending bills. The appropriations process is a challenge in good years. Amid broader battles over taxes and whether to expand or shrink the government, it has been derailed in recent ones.

Boehner, for example, sided with the GOP’s tea party wing in 2013 and saddled Rogers with a budget outline that pleased conservatives but shortchanged domestic programs and doomed appropriators to failure last summer. Rogers could only watch as tea party forces steamrolled House leaders into a government shutdown last fall.

What eventually came out of it was a two-year budget deal that paved the way for Rogers and Mikulski to get the process back on track. In January they accomplished several months’ worth of work in a few weeks, negotiating and winning passage of a $1.1 trillion catchall spending bill for 2014.

Jim Dyer, a former Republican staff director for Senate appropriators, said Rogers and Mikulski were under a lot of pressure to reach a deal that could pass both houses. “If they had failed, people would have thrown the last shovel of dirt on appropriations,” Dyer said. “Now, what you have is ... the only show in town currently working well — and these two people deserve the credit.”